Table of ContentsMeasures of central Tendency

2.1 Introduction2.1.1 Meaning2.1.2 Objectives Of Averaging2.1.3. Characteristics Of A Good Average2.1.4. Types Of Averages

2.2. Arithmetic mean2.2.1. Calculation Of Arithmetic Mean2.2.2 Mathematical Properties Of Arithmetic Mean

Measures of central Tendency

Mathematics & Statistics for Business (Statistics)

Lesson: Measures of central TendencyLesson Developer: Madhu Gupta

College/Department: Janki Devi Memorial College of Commerce, University ofDelhi

2.1 Introduction

Fig. 2.1 Balance depicting measure of central tendency Source: http://images.google.co.in/images?

hl=en&source=hp&q=balance&btnG=Search+Images&gbv=2&aq=f&aqi=&aql=&oq=&gs_rfai=

After collection, classification and presentation of data we need to analyze and interpret the same. Thereare various numerical measures which describe the inherent characteristics of collected data.(see web link1). The first such measure is “measure of central tendency” or ‘average’ which gives us a single valuewhich describes the characteristics of the entire group and facilitates easy expression, reference andinterpretation. It reduces the complex mass of data into a single figure, which is “gist” if not the substanceof the series.

Value addition: Did you Know?

Descriptive statistics

A frequency distribution or data set can be fully described byfour main characteristics (called descriptive statistics). Theseare :

1. Measure of central tendency – measures a single valuearound which all or most of the values in the data set clusters.

2. Measure of dispersion – measures the extent to whichindividual values are spread.

http://www.techshout.com/software/2006/16/microsoft­excel­hit­by­new­vulnerability/

3. Measure of skewness – measures the extent of departureof individual values from symmetrical distribution. It describesthe shape of the distribution. A distribution can be positivelyskewed, symmetrical or negatively skewed.

4. Measure of kurtosis– measures the height or peakednessof the distribution as compared to normal distribution.

2.1.1 MeaningMeasure of central tendency, also known as average, is one of the most widely used statistical device. Itis that value of the distribution where most of the items of the series tend to cluster. Average is computedto obtain a single value which can represent a whole series of observations. A measure of central tendency,by condensing a mass of data in one single value, enables us to get a bird’s eye view of the entire data andmake comparisons. For example if the marks obtained by 1000 students of college A and 1000 students ofcollege B are before us, it may not be possible for us to draw any inference from them or to compare thetwo series relating to these marks readily. On the other hand, comparison would become very easy if weget a single value to represent the marks of all the 1000 students of college A and another single value torepresent the marks of all the 1000 students of college B. An average is used to represent a whole seriesand as such, its value always lies between the minimum and maximum value and generally it is located inthe centre or middle of the distribution.

Value addition: Did you Know?

The word average is very commonly used in day to dayconversations, for example, we often talk about pass percentageof a university or college, average height, average income ofemployees in an industry, life expectancy of an average male orfemale etc. In the world of business and finance, BSE SENSEX,NIFTY, wholesale and consumer price index, per capita income,percentage of growth in GDP, inflation, population, & rate ofreturn on investment etc. are some of the commonly usedaverages.

Averages also appears daily on T.V., in the newspaper andother journals. Here are some interesting facts gathered fromvarious published media:

1. Average daily footfalls in Ambience Mall in Gurgaon are59356.

2. Average loan size of urban poor has increased from Rs.5000in 2008 to Rs.10000 in 2009.

3. Average annual package offered by international banks to B­School Indian graduates is Rs. 15 lakhs.

4. Average net worth of top 25 Indian billionaires is Rs.8152.8millions Dollar.

5. Life expectancy of an average Indian is only 66.09 years ascompared to 77.5 years of an

American.

6 In 2009, fertility rate in India was 2.68 children born/woman.

7 Sensex was 17933.14 on 9th April 2010.

8 In March 2010, food article price index rose to 17.7%.

Value addition: Interesting Facts!

Some interesting facts about average from everyday life:

Cats average 16 hours of sleep a day, more than any other mammal.

Every square inch of the human body has an average of 32 million bacteriaon it.

In 1900 the average age at death in the US was 47.

On an average a woman speaks 7,000 words per day; Men manage withjust over 2000.

On an average, 42,000 balls are used and 650 matches are played at theannual Wimbledon

tennis tournament.

On an average, right­handed people live 9 years longer than their left­handed counterparts.

Smokers are likely to die on an average six and a half years earlier thannon­smokers.

The average person drinks about 16, 000 gallons of water in a lifetime.

The average person walks the equivalent of twice around the world in alifetime.

Source ;http://www.corsinet.com/trivia/average.html

Picture source: http://images.google.co.in/images?hl=en&gbv=2&tbs=isch:1&q=cartoons+pictures&sa=N&start=40&ndsp=20

Average value should be such where most of the items of the series tend to cluster. In bell shapeddistributions, this is true and frequencies tend to cluster around the average. But in case of J­shaped(skewed) and U­ shaped distributions this is not true and frequencies do not cluster around average. Inthese cases proper care must be taken while interpreting its value.

The averages are also called measures of location because they determine the position of thedistribution on the axis of the variable.

fig. 2.3 Centre of location of three distributions

2.1.2 Objectives Of AveragingThere are two main objectives of averaging:

1. To get a single values to represent all the individual values in the series

Average gives a bird’s eye view of the huge mass of data, which are ordinarily not intelligible. It setsaside the unnecessary details of the data and aid the human mind in grasping the true significance of largeaggregates of facts and measurements. Thus, the questions like ‘how cold was Delhi in January this year’,or ‘what was the electricity consumption in June in a city’ may be answered by a single figure of averagetemperature in January and average electricity consumption of the city in June. It is not necessary to givehuge unwieldy set of numerical data of temperatures and electricity consumptions of different days.

Fig 2.4­ Average temperature of New­Delhi,2009

Source­ http://www.bbc.co.uk/weather/world/city_guides/results.shtml?tt=TT002240

Value addition: Misconceptions

Average is a single value which represents a whole series ofobservations. However It does not mean all values of thedistribution are uniform and are equal to average. Averagealways lies somewhere between the lowest and the highestvalues and proper care should be taken in interpreting its value

In this regard there is an interesting old story of a person who,along with his family of ten members, was going to a neighboringvillage. In between they had to cross a river whose averagedepth was understood to be 4 feet. He calculated the averageheight (mean) of the members of his family and found it to be 4feet 8 inches. Since average height was more than the averagedepth of the river, he ordered his family to cross the river. Fourmembers of his family drowned while crossing. He was verysurprised and checked his calculations again which came out tobe correct. He could not understand where he made the mistakeand finally remarked­

‘Ausat jyon ka tyon, kunba doobakyon?’

( the average is the same , why has thefamily drowned )

In fact, average depth and average height does not meanuniform depth and uniform height. Individual values may be lessor more than the average value.

Fig.Source­http://www.1st­art­gallery.com/(after)­Le­Moyne,­Jacques­(de­Morgues)/A­Family­Crossing­A­River.html

2. To facilitate comparison

Averages are very helpful for making comparative studies. For example, if the average of the marks ofthe students of college A is 72 % and that of college B is 78 %, we may say performance of the students ofcollege B is better than that of college A. This would not be possible if we had two full series of marks ofindividual students of the two colleges. But this certainly doesn’t mean that all students of college B arebetter than that of college A In both the colleges there may be certain students who must have scored lessthan the average marks and similarly there will be certain students who must have scored more than theaverage marks. We have to be very careful while making such comparisons and must consider otherfactors such as dispersion before drawing any inferences.

2.1.3. Characteristics Of A Good AverageThere are different types of averages, and, the question is, which one should be used. Since it is to be usedto represent all the values in a series it should satisfy certain properties.

First is should be easy to understand and calculate so that a person of ordinary intelligence easilyunderstands it. Others wise its use will be confined to a limited number of persons. However convenienceof computation should not be sought at the expense of accuracy.

Source­http://images.google.co.in/images?hl=en&source=hp&q=statistics&btnG=Search+Images&gbv=2&aq=f&aqi=&aql=&oq=&gs_rfai

Second, it should be rigidly defined so that it is subject to one and only one interpretation. An averageshould be a precise and definite value. The user may not introduce any bias i.e. it is not left to the mereestimation of the observer, because in that case it will not be a true representative of the series. It will bepreferable if it can be defined by an algebraic formula so that if different people compute the average fromthe same figures, they all get the same value.

Third, it should be based on all the item of the series. If some of values of the series are not taken intoaccount in computing an average, it cannot be said to be a true representative of the whole series.

Fourth, through an average should be based on all the items of the series, it should not be unduly affectedby extreme observations. If one or two very small or very large items are able to influence very much thevalue of the average then such average will not be the true representative of the series

Fig 2.6 A few bigger items influencing the average

Last, it should have sampling stability and should be capable of further algebraic treatment. For thisaverage should have a well built mathematical model so that different persons calculating the sameaverage from the different samples of the same population should get approximately the same value. Nodoubt, averages of different samples are rarely the same, but one form of average may show muchgreater differences in the values of two samples than another. Of the two, that one in better in which thisdifference, technically called fluctuations’ of sampling is less, otherwise it will have limited application and

utility.

It should be noted that no single average satisfies all the above properties and is sailable for all practicalpurposes. A suitable selection of an average depends on the nature of data and purpose of enquiry.

2.1.4. Types Of AveragesThe different averages which we will study here are arithmetic mean, median, mode, geometric mean andharmonic mean. These averages are often categorized as mathematical averages or positionalaverages. The averages like arithmetic mean geometric mean and harmonic mean are calledmathematical averages since they are computed with the help of a well built mathematical formula and arebased on each and every value of the variable. Median and mode are called positional averages becausetheir values are determined on the basis of their position in the distribution and not by the size of theitems.

2.2. Arithmetic meanArithmetic mean is the most important and widely used average. Most of the time, when we refer to theaverage of something, we are talking about Arithmetic mean. This is true in cases such as average income,average marks, and average temperature of a city. It is equal to the total of all the values in a seriesdivided by the number of items in that series.

2.2.1. Calculation Of Arithmetic Mean

Direct method.

Short cut method

When the values are large and the number of items is high, a short – cut method may be used to reducethe calculations. Under this method, one value is taken as an arbitrary origin or assumed mean, deviationsof individual values are taken from that assumed meaan and then arithmetic mean is calculated by usingthe following formula:

Solution: Calculation of mean

Salary (in 000Rs)

No. ofWorkers

f X X­A (A=63) fd

X F d

60 2 120 ­3 ­661 9 549 ­2 ­1862 10 620 ­1 ­1063 15 945 0 064 7 448 1 765 4 260 2 866 3 198 3 9

f = 50 fx = 3140 ∑fd = ­10

C. Continuous series

Direct method

First, mid­point of each class is taken as the representative figure of that particular class. Then wemultiply each mid­point by the corresponding frequency and apply the following formula to find out thearithmetic mean:

Short­cut method

In this method, deviations of mid­points from an arbitrary origin are taken. The formula applied tocalculate mean is

Step­deviations method

This is an extension of the short cut method used for further simplification. Where figures of deviations aredivisible by a common factor, we reduce them by dividing them by that common factor and then multiplytheir sum by the same common factor. Thus

C is common factor

Illustration: Following is the data of overtime hours worked by 100 employees of GPL ltd.. You are

required to calculate arithmetic mean from the following data by using direct, short­cut and step deviationmethod.

Overtime (hrs) 0­10 10­20 20­30 30­40 40­50

No. of employers 15 25 30 20 10

Solution:

The result obtained in case of continuous series will be only approximate of the actual average. This isbecause we do not know every data point in the sample. We assumed that every value in a class was equalto its mid point.

2.2.2 Mathematical Properties Of Arithmetic MeanArithmetic mean has some very important mathematical properties which are as follows:­

1. if the number of items and their aritmetic mean are both know the aggregate or the sum of items can beobtained by multiplying the average by the number of items.

this property of mean in known as "Total value" property and is not possessed by any other measure ofcentral tendency.

Illustration: The average monthly sales for the first 11 months of the year on respectof a certain salesman were rs 24000 but due to his illness during the last month theaverage sales fr the whole year came down to Rs. 22,750. What was the value of hissales during the last month.

Solution:

or total sales = Average sales x no. of months.Now let the sales during the last month br K.

Then the total sales of 11 month = 24000 x 11 = 264000 Rs and total sales of 12 months (wholeyear) = 264000 + K.

Average of the whole year = 22,750 (given)

Or 273000 = 264000 + K

Or k = 9000 Rs

sales of last month are Rs 9000

Illustration: The mean of 40 observations wae 160. It was detected on rechecking that twovalues 105 and 180 were wrongly copied as 125 and 120 for the computation of mean. Findthe correct mean.

Soultions:

Here,

Incorrect sum of 40 observations = 160 x 40 = 6400

Correct sum of 4 observations = 6400 ­ 125 ­ 120 + 105 +108 = 6440

Correct mean =

2. The sum of the deviations of the individual items from their arithmetic mean is always zero. i. e.

For example, the arithmetic mean of 15, 20, 25, 30 and 35 is 25. if the difference of each of these items iscalculated it whould be ­10, ­5, 0, +5 and + 10. Their total is zero.

Due to this property the mean is also called point of balance i.e. at this point (mean) sum of the positivedeviations from mean is equal to the sum of the negative deviations from mean.

Fig 2.10 Arithmetic mean as a balance point

3. The sum the squared deviations of the items from arithmetic mean is minimum i.e. it is less that thesum of the squared deviations of the items from any other value.